Our work considers rigid body dynamics [1]}. Translations, rotations and their compositions are the primitive motions of such systems. This is a broad class of dynamical systems, used ubiquitously in artificial intelligence and optimal control [2]}, [3]}, [4]}. For example, the motion of a planar single pendulum, which is a two-dimensional non-linear dynamical system, is characterized by rotation with regard to its pivot. For another example, the motion of a humanoid, which is a high-dimensional system, is characterized by a composition of multiple limb rotations and translation of center of mass. An important feature of these primitive motions is that they are elements of the corresponding groups, or more precisely, Lie groups.
